DHAKA, Bangladesh—Hannah Jones, Nike Inc.'s head of sustainable business, had been lecturing colleagues for years about the dangers of manufacturing in Bangladesh. Yes, the country featured some of the cheapest factories in the world, she argued, but the athletic-gear maker could ill afford another public pasting over its labor practices.

Her counterparts in the production division, charged with squeezing costs, countered that they should all visit the place together and then decide. So one day last year, five of them...